Volumetric imaging of holographic optical traps

ABSTRACT

A method and system for manipulating object using a three dimensional optical trap configuration. By use of selected hologram on optical strap can be configured as a preselected three dimensional configuration for a variety of complex uses. The system can include various optical train components, such as partially transmissive mirrors and Keplerian telescope components to provide advantageously three dimensional optical traps.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

This application claims the benefit under 35 USC 119(e) of U.S.Application No. 60/852,252, filed Oct. 17, 2006, incorporated herein byreference in its entirety.

The United States Government has certain rights in this inventionpursuant to a grant from the National Science Foundation through grantnumber DMR-0451589.

This invention is directed toward volumetric imaging of holographicoptical traps. More particularly, the invention is directed to a methodand system for creating arbitrary pre-selected three-dimensional (3D)configurations of optical traps having individually specified opticalcharacteristics. Holographic techniques are used to modify individualtrap wavefronts to establish pre-selected 3D structures havingpredetermined properties and are positionable independently in threedimensional space to carry out a variety of commercially useful tasks.

BACKGROUND OF THE INVENTION

There is a well developed technology of using single light beams to forman optical trap which applies optical forces from the focused beam oflight to confine an object to a particular location in space. Theseoptical traps, or optical tweezers, have enabled fine scale manipulationof objects for a variety of commercial purposes. In addition, linetraps, or extended optical tweezers, have been created which act as aone dimensional potential energy landscape for manipulating mesoscopicobjects. Such line traps can be used to rapidly screen interactionsbetween colloidal and biological particles which find uses in biologicalresearch, medical diagnostics and drug discovery. However, theseapplications require methods of manipulation for projecting line trapswith precisely defined characteristics which prevent their use insituations with high performance demands. Further, the low degrees offreedom and facility of use for such line traps reduces the ease of useand limits the types of uses available.

SUMMARY OF THE INVENTION

The facility and range of applications of optical traps is greatlyexpanded by the method and system of the invention in which 3D intensitydistributions are created by holography. These 3D representations arecreated by holographically translating optical traps through an opticaltrain's focal plane and acquiring a stack of two dimensional images inthe process. Shape phase holography is used to create a pre-selected 3Dintensity distribution which has substantial degrees of freedom tomanipulate any variety of object or mass for any task.

Various aspects of the invention are described hereinafter; and theseand other improvements are described in greater detail below, includingthe drawings described in the following section.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an optical train for performing a method of theinvention;

FIG. 2A illustrates a particular optical condition with z<0 for anobjective lens in the system of FIG. 1; FIG. 2B illustrates the opticalcondition for z=0 for the objective lens of FIG. 1 and FIG. 2Cillustrates the optical condition for z>0 for the objective lens of FIG.1;

FIG. 3A illustrates a 3D reconstruction of an optical tweezerpropagating along the z axis; FIG. 3B illustrates a cross-section ofFIG. 3A along an xy plane; FIG. 3C illustrates a cross-section of FIG.3A along a yz plane; FIG. 3D illustrates a cross-section of FIG. 3Aalong an xz plane; FIG. 3E illustrates a volumetric reconstruction of 35optical tweezers arranged in a body-centered cubic lattice of the typeshown in FIG. 3F;

FIG. 4A illustrates a 3D reconstruction of a cylindrical lens lineoptical tweezer; FIG. 4B illustrates a cross-section of FIG. 4A along anxy plane; FIG. 4C illustrates a cross-section of FIG. 4A along a yzplane; and FIG. 4D illustrates a cross-section of FIG. 4A along an xzplane; and

FIG. 5A illustrates a 3D reconstruction of a holographic optical trapfeaturing diffraction-limited convergence to a single focal plane; FIG.5B illustrates a cross-section of FIG. 5A along a xy plane; FIG. 5Cillustrates a cross-section of FIG. 5A along a yz plane; and FIG. 5Dillustrates a cross-section of FIG. 5A along an xz plane.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

An optical system for performing methods of the invention is illustratedgenerally at 10 in FIG. 1. A beam of light 20 is output from afrequency-doubled solid-state laser 30, preferably a Coherent Verdisystem operating at a wavelength of λ=532 nm. The beam of light 20 isdirected to an input pupil 40 of a high-numerical-aperture objectivelens 50, preferably a Nikon 100 x Plan Apo, NA 1.4, oil immersion systemthat focuses the beam of light 20 into an optical trap (not shown). Thebeam of light 20 is imprinted with a phase-only hologram by acomputer-addressed liquid-crystal spatial light modulator 60 (“SLM 60”),preferably a Hamamatsu X8267 PPM disposed in a plane conjugate to theobjective lens' 50 input plane. Computer 95 executes conventionalcomputer software to generate the appropriate hologram using the SLM 60.As a result, the light field, ψ(r), in the objective lens' 50 focalplane is related to the field ψ/(ρ) in the plane of the SLM 60 by theFraunhofer transform, $\begin{matrix}{{{\psi(r)} = {{- \frac{i}{\lambda\quad f}}{\int_{\Omega}{{\psi(\rho)}{\exp( {{- i}\frac{2\quad\pi}{\lambda\quad f}{r \cdot \rho}} )}\quad{\mathbb{d}{\,^{2}\rho}}}}}},} & (1)\end{matrix}$where ƒ is the objective's focal length, where Ω is the optical train'saperture, and where we have dropped irrelevant phase factors. Assumingthat the beam of light 20 illuminates the SLM 60 with a radiallysymmetric amplitude profile, u(ρ), and uniform phase, the field in theSLM's plane may be written as,ψ(ρ)=u(ρ)exp(iφ(ρ)),  (2)where φ(ρ) is the real-valued phase profile imprinted on the beam oflight 20 by the SLM 60. The SLM 60 in our preferred form of the system10 imposes phase shifts between 0 and 2π radians at each pixel of a768×768 array. This two-dimensional phase array can be used to project acomputer-generated phase-only hologram, φ(ρ), designed to transform thesingle optical tweezer into any desired three-dimensional configurationof optical traps, each with individually specified intensities andwavefront properties.

Ordinarily, the pattern of holographic optical traps would be put to useby projecting it into a fluid-borne sample mounted in the objectivelens' 50 focal plane. To characterize the light field, we instead mounta front-surface mirror 70 in the sample plane. This mirror 70 reflectsthe trapping light back into the objective lens 50, which transmitsimages of the traps through the partially reflecting mirror 70 to acharge-coupled device (CCD) camera 80, preferably a NEC TI-324AII. Inour implementation, the objective lens 50, the camera 80 and cameraeyepiece (not shown), are mounted in a conventional optical microscope(not shown) and which is preferably a Nikon TE-2000U.

Three-dimensional reconstructions of the optical traps' intensitydistribution can be obtained by translating the mirror 70 relative tothe objective lens 50. Equivalently, the traps can be translatedrelative to the mirror 70 by superimposing the parabolic phase function,$\begin{matrix}{{{\varphi_{z}(\rho)} = {- \frac{\pi\quad\rho^{2}z}{\lambda\quad f^{2}}}},} & (3)\end{matrix}$onto the hologram φ₀(ρ) encoding a particular pattern of traps. Thecombined hologram, φ₀(ρ)=φ₀(ρ)+φ_(z)(ρ) mod 2π, projects the samepattern of traps as φ₀(ρ) but with each trap translated by −z alongoptical axis 90 of the system 10. The resulting image obtained from thereflected light represents a cross-section of the original trappingintensity at distance z from the focal plane of the objective lens 50.Translating the traps under software control by computer 95 isparticularly convenient because it minimizes changes in the opticaltrain's properties due to mechanical motion and facilitates moreaccurate displacements along the optical axis 90. Images obtained ateach value of z are stacked up to yield a complete volumetricrepresentation of the intensity distribution.

As shown schematically in FIGS. 2A-2C, the objective lens 50 capturesessentially all of the reflected light for z<0. For z>0, however, theoutermost rays of the converging trap are cut off by the objective lens'50 output pupil 105, and the contrast is reduced accordingly. This couldbe corrected by multiplying the measured intensity field by a factorproportional to z for z>0. The appropriate factor, however, is difficultto determine accurately, so we present only unaltered results.

FIG. 3A shows a conventional optical tweezer 100 reconstructed in themanner described hereinbefore and displayed as an isointensity surfaceat 5 percent peak intensity and in three cross-sections (FIGS. 3B-3D).The representation in FIG. 3A is useful for showing the overallstructure of the converging light, and the cross-sections of FIGS. 3B-3Dprovide an impression of the three dimensional light field that willconfine an optically trapped object. The angle of convergence of 63° inimmersion oil obtained from these data is consistent with an overallnumerical aperture of 1.4. The radius of sharpest focus, r_(min)≈0.21μm, is consistent with diffraction-limited focusing of the beam of light20.

These results highlight two additional aspects of this reconstructiontechnique. The objective lens 50 is designed to correct for sphericalaberration when the beam of light 20 passing through water is refractedby a glass coverslip. Without this additional refraction, the projectedoptical trap 100 actually is degraded by roughly 20λ of sphericalaberration, introduced by the objective lens 50. This reduces theapparent numerical aperture and also extends the trap's focus along thez axis. The trap's effective numerical aperture in water would beroughly 1.2. The effect of spherical aberration can be approximatelycorrected by pre-distorting the beam of light 20 with the additionalphase profile, $\begin{matrix}{{{\varphi_{a}(\rho)} = {\frac{a}{\sqrt{2}}( {{6\quad x^{4}} - {6\quad x^{2}} + 1} )}},} & (4)\end{matrix}$the Zernike polynomial describing spherical aberration. The radius, x,is measured as a fraction of the optical train aperture, and thecoefficient α is measured in wavelengths of light. This procedure isused to correct for small amount of aberration present in practicaloptical trapping systems to optimize their performance.

This correction was applied to an array 110 of 35 optical tweezers shownas a three-dimensional reconstruction in FIG. 3E. These optical traps100 are arranged in a three-dimensional body-centered cubic (BCC)lattice 115 shown in FIG. 3F with a 10.8 μm lattice constant. Withoutcorrecting for spherical aberration, these traps 100 would blend intoeach other along the optical axis 90. With correction, their axialintensity gradients are clearly resolved. This accounts for holographictraps' ability to organize objects along the optical axis.

Correcting for aberrations reduces the range of displacements, z, thatcan be imaged. Combining φ_(α)(ρ) with φ_(z)(ρ) and φ₀(ρ) increasesgradients in φ(ρ), particularly for larger values of ρ near the edges ofthe diffraction optical element. Diffraction efficiency falls offrapidly when |∇φ(ρ)| exceeds 2π/Δρ, the maximum phase gradient that canbe encoded on the SLM 60 with pixel size Δρ. This problem is exacerbatedwhen φ₀(ρ) itself has large gradients. In a preferred embodiment morecomplex trapping patterns without aberration are prepared. Inparticular, we use uncorrected volumetric imaging to illustrate thecomparative advantages of the extended optical traps 100.

The extended optical traps 100 have been projected in a time-sharedsense by rapidly scanning a conventional optical tweezer along thetrap's intended contour. A scanned trap has optical characteristics asgood as a point-like optical tweezer, and an effective potential energywell that can be tailored by adjusting the instantaneous scanning rateKinematic effects due to the trap's motion can be minimized by scanningrapidly enough. For some applications, however, continuous illuminationor the simplicity of an optical train with no scanning capabilities canbe desirable.

Continuously illuminated line traps have been created by expanding anoptical tweezer 125 along one direction (see FIG. 4A). This can beachieved, for example, by introducing a cylindrical lens component suchas by element 130 (see FIG. 1) into the objective's input plane.Equivalently, a cylindrical-lens line tweezer can be implemented byencoding the function φ_(c)(ρ)=πz₀ρ_(x) ²/(λƒ²) on the SLM 60. Theresult, shown in FIGS. 4A-4D appears best useful in the plane of bestfocus, z=z_(o), with the point-like tweezer having been extended to aline with nearly parabolic intensity and a nearly Gaussian phaseprofile. The three-dimensional reconstruction, however, reveals that thecylindrical lens component merely introduces a large amount ofastigmatism into the beam of light 20, creating a second focal lineperpendicular to the first. This is problematic for some applicationsbecause the astigmatic beam's axial intensity gradients are far weakerthan a conventional optical tweezer's. Consequently, cylindrical-lensline traps typically cannot localize objects against radiation pressurealong the optical axis 90.

Replacing the single cylindrical lens with a cylindrical Kepleriantelescope for the element 130 eliminates the astigmatism and thuscreates a stable three-dimensional optical trap. Similarly, using theobjective lens 50 to focus two interfering beams creates aninterferometric optical trap capable of three-dimensional trapping.These approaches, however, offer little control over the extended traps'intensity profiles, and neither affords control over the phase profile.

Shape-phase holography provides absolute control over both the amplitudeand phase profiles of an extended form of the optical trap 100 at theexpense of diffraction efficiency. It also yields traps with optimizedaxial intensity gradients, suitable for three-dimensional trapping. Ifthe line trap is characterized by an amplitude profile ũ(ρ_(x)) and aphase profile {tilde over (p)}(ρ_(x)) along the {circumflex over(ρ)}_(x) direction in the objective's focal plane, then the field in theSLM plane is given from Eq. (1) as,ψ(ρ)=u(ρ_(x))exp(ip(ρ_(x))),  (5)where the phase p(ρ_(x)) is adjusted so that u(ρ_(x))≧0. Shape-phaseholography implements this one-dimensional complex wavefront profile asa two-dimensional phase-only hologram, $\begin{matrix}{{\varphi(\rho)} = \{ {\begin{matrix}{{p( \rho_{x} )},} & {{S(\rho)} = 1} \\{{q(\rho)},} & {{S(\rho)} = 0}\end{matrix},} } & (6)\end{matrix}$where the shape function S(ρ) allocates a number of pixels along the rowρ_(y) proportional to u(ρ_(x)). One particularly effective choice is forS(ρ) to select pixels randomly along each row in the appropriaterelative numbers. The unassigned pixels then are given values q(ρ) thatredirect the excess light away from the intended line. Typical resultsare presented in FIG. 5A.

Unlike the cylindrical-lens trap, the holographic line trap 130 in FIGS.5A-5D focuses as a conical wedge to a single diffraction-limited line inthe objective's focal plane. Consequently, its transverse angle ofconvergence is comparable to that of an optimized point trap. This meansthat the holographic line trap 120 has comparably strong axial intensitygradients, which explains its ability to trap objects stably againstradiation pressure in the z direction.

The line trap's transverse convergence does not depend strongly on thechoice of intensity profile along the line. Its three-dimensionalintensity distribution, however, is very sensitive to the phase profilealong the line. Abrupt phase changes cause intensity fluctuationsthrough Gibbs phenomenon. Smoother variations do not affect theintensity profile along the line, but can substantially restructure thebeam. The line trap 120 created by the cylindrical lens element 130 forexample, has a parabolic phase profile. Inserting this choice into Eq.(2) and calculating the associated shape-phase hologram with Eqs. (1)and (6) yields the same cylindrical lens phase profile. This observationopens the door to applications in which the phase profile along a linecan be tuned to create a desired three-dimensional intensitydistribution, or in which the measured three-dimensional intensitydistribution can be used to assess the phase profile along the line.These applications will be discussed elsewhere.

The foregoing description of embodiments of the present invention havebeen presented for purposes of illustration and description. It is notintended to be exhaustive or to limit the present invention to theprecise form disclosed, and modifications and variations are possible inlight of the above teachings or may be acquired from practice of thepresent invention. The embodiments were chosen and described in order toexplain the principles of the present invention and its practicalapplication to enable one skilled in the art to utilize the presentinvention in various embodiments, and with various modifications, as aresuited to the particular use contemplated.

1. A method of manipulating objects using three dimensional opticaltraps, comprising the steps of: providing an optical train; providing abeam of light to the optical train; applying a predetermined hologram tothe beam of light to form an optical trap; further applying a phase onlyhologram to establish a predetermined three dimensional optical trapconfiguration; and using the three dimensional optical trap tomanipulate an object.
 2. The method as defined in claim 1 wherein thephase only hologram is applied to selected ones of a plurality of theoptical trap, the step including modifying wavefronts of the opticaltrap.
 3. The method as defined in claim 1 wherein the steps of applyinga phase only hologram includes having the optical train with a mirroroperating on the beam of light to characterize a light field of theoptical trap.
 4. The method as defined in claim 3 wherein the mirror istranslated relative to an objective lens of the optical train to performa three dimensional reconstruction of intensity of the optical trap. 5.The method as defined in claim 3 wherein the optical trap is translatedrelative to a fixed form of the mirror by including in the hologram aparabolic phase function.
 6. The method as defined in claim 5 whereinthe parabolic phase function comprises,${{\varphi_{z}(\rho)} = {- \frac{\pi\quad\rho^{2}z}{\lambda\quad f^{2}}}},$7. The method as defined in claim 5 wherein computer software executedby a computer creates the hologram for translating the optical trap. 8.The method as defined in claim 1 further including the step of passingthe beam of light through an objective lens of the optical train andapplying a phase profile to the optical trap, thereby enabling theplurality of optical traps to organize selected objects along an opticalaxis of the optical train.
 9. The method as defined in claim 8 whereinthe phase profile comprises,${{\varphi_{a}(\rho)} = {\frac{a}{\sqrt{2}}( {{6\quad x^{4}} - {6\quad x^{2}} + 1} )}},$thereby correcting for spherical aberration.
 10. The method as definedin claim 1 further including the step of forming a single optical trapand rapidly scanning the optical trap along the predetermined threedimensional optical trap configuration.
 11. The method as defined inclaim 1 further including the step of introducing an objective lens inthe optical train and introducing a cylindrical lens component into aninput plane of the objective lens.
 12. The method as defined in claim 1wherein the step of applying a predetermined hologram includesintroducing a cylindrical-lens line tweezer component into the hologram.13. The method as defined in claim 12 wherein the predetermined hologramincludes the function φ_(c)(ρ)=φ_(c)(ρ)=πz₀ρ_(x) ²/(λƒ²).
 14. The methodas defined in claim 1 further including in the optical train a Kepleriantelescope for eliminating astigmatism, thereby creating a stable threedimensional optical trap.
 15. The method as defined in claim 1 whereinthe optical trap comprises a hologram constructed of, a. an amplitudeprofile ũ(ρ_(x)). b. a phase profile {tilde over (p)}(ρ_(x)) with anobjective focal plane direction {circumflex over (ρ)}_(x).
 16. Themethod as defined in claim 16 wherein the optical train includes an SLMhaving an associated plane and the field in the associated plane isgiven by,ψ(ρ)=u(ρ_(x))exp(ip(ρ_(x))).
 17. The method as defined in claim 1wherein the phase only hologram comprises,${\varphi(\rho)} = \{ {\begin{matrix}{{p( \rho_{x} )},} & {{S(\rho)} = 1} \\{{q(\rho)},} & {{S(\rho)} = 0}\end{matrix}.} $
 18. A system for manipulating an object usingthree dimensional optical traps, comprising: an optical train; a sourceof light with the light input to the optical train; a spatial lightmodulator; and a computer having a computer program to control thespatial light modulator to generate a phase controlled hologram, thehologram providing a predetermined three-dimensional configuration ofoptical traps for manipulating the object.
 19. The system as defined inclaim 18 wherein the hologram comprises a phase only hologram.
 20. Thesystem as defined in claim 18 further including at least one of (1) amirror in the optical train for operating on the light to characterize alight field of the optical traps, (2) the mirror being translated alongthe optical train relative to the objective lens, to perform a threedimensional reconstruction of light intensity of the optical traps and(3) a modified parabolic phase function hologram to translate theoptical traps relative to a fixed form of the mirror.